Tuesday, March 07, 2006

1 in 649,740

Last night I got dealt an ace and ten in clubs. The flop: K Q J in clubs. I got a royal flush on the flop. This site says the odds of getting a royal flush in Texas Hold 'Em are 1 in 649,740. This means that you have to play 324,870 hands before you even have a 50% chance of having it. This site reports that the odds of getting struck by lightning are 576,000 to 1 (so I better watch out), and the odds of becoming President or getting killed by a falling part from an airplane are 10,000,000 to 1, so I'm not sure if I want to get 14 more royal flushes in my life. And the odds are about the same as those for drowning in a bathtub.

6 comments:

Heh, see this is what I mean about statistics! The odds are actually better than that.

See in those 300,000 times you are not just interested in getting the royal flush once- you could get it twice, three times, or every single time. So while it is accurate to say that you need to play that that many times to get a 50% chance of achieving the royal flush exactly once, you need to play it less to have a 50% probablity of getting at least once.

... apart from that's wrong. You actually need MORE than 300,000 times, because the situation is a little more complex. My argument is incorrect, however.

Basically, you have a probability of achieving something of 1/n. The probablity of not doing this is 1-1/n. So the probablity of achieving our event is 1-1-1/n. Now how about achieving it once if we repeat our experiment twice- well it will be 1 minus the probailiy of not getting it twice- or 1-(1-1/n)^2.

So to get our answer to we must solve

1-(1-1/n)^m=1/2 where m is the amount of times we are performing the event.

We get

(1-1/n)^m=1/2

taking logs

mlog(1-1/n)=log(1/2)m=(log(1/2))/log(1-1/n)

In this case, the answer is you need to play 450365.1026 hands to have a 50% probablity of getting at least one royal flush....

Thanks. I never took statistics, because it has the same effect on me as NyQuil, but without the good feelings.

And actually I did the math myself on the original statistic and it turns out that those are the odds of getting a royal flush on the first five cards (about 0.00015%), but in Texas Hold 'Em you get two more cards to work with--i.e. you get 7 cards to make the best 5-card hand possible. I'm not exactly sure how that changes the math.

heh, not to fill this comments page with me, but I think the effect of having seven cards to get a 5 card combination basically multiplies the probability by 7C2 (7!/(5!2!) where n!=n*(n-1)*(n-2)..) As to doing the relevant calcualtions from here on... I can't be bothered. I'm actually not too sure about all this, having not done pure probability for two years, instead studying statistics.

I think I got as much as I could. The blinds were low at the time, and I was pretty sure they both had a straight and thought we'd split the pot. I bid 60 at the end, netting 120 from the other two guys. I think if I would have bid any more, then one of them would have folded, and if I bet 120 they both would have folded.

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